The present study is consisting of two main parts. In the first part, some numerical meshless methods are introduced for analyzing space and time-dependent engineering mechanic problems using local exponential basis functions (EBFs). In these methods a specific approach has been used for space discretization and the solution process through the time is achieved by a time marching method. In each time step, the solution is approximated as a series of spatial or spatial-temporal exponential basis functions with unknown coefficients. The coefficients are found in terms of nodal values through a fitting process so that from one hand the boundary/initial conditions are satisfied and from the other hand the continuity of the solution in time is acceptably obtained. After that, the continuity equation needs to be satisfied for the degrees of freedom (DOFs) of each cloud. Going through this process for the entire domain, the final system of equations is obtained which leads to finding the unknown nodal values. The main innovation of the present study in this part is the way of satisfying the continuity condition in the time domain which is the basic difference between mentioned methods. Stability and accuracy of these methods for solving unsteady heat conduction problems, scalar wave propagation and elastic wave propagation are investigated and acceptable results are achieved. The examples are including problems with regular and irregular domains with Dirichlet and Neumann boundary conditions as well as problems with singularities. In the second part, a local meshless method, based on the last introduced method of the second chapter, is presented to solve two-dimentional Navier-Stokes equations for incompressible viscous fluids. The computational frame is Lagrangian-Eulerian, which means that the fluid equations are expressed in Lagrangian formulation, containing velocity and acceleration of the particles, but the fluid description is based on Eulerian viewpoint which means that the computation is performed on a fixed grid of points. With such a Lagrangian-Eulerian formulation there is no need to update the locations of the points in each time step, so that the computational speed increases while its cost decreases. It is worth mentioning the formulation of proposed method in this part, provide the ability to calculate the velocity field of the fluid, independent of pressure field, vorticity or stream function. That is another innovation of present investigation. After determination of the velocity field of the fluid, the nodal values of the pressure gradient can be calculated, therefore it is possible to determine the pressure field as well. Stability and accuracy of the presented method are demonstrated through the solution of a lid driven cavity at different Reynolds number. Good agreement with reference numerical solutions is observed. Key words : Numerical meshless method, Space and time-dependent problems, Local exponential basis functions, Continuity in time, Stability and accuracy, Incompressible viscous fluid, Lagrangian-Eulerian formulation, Reynolds number.